Capacitors and Capacitance
A capacitor is a two-terminal electrical device that can store energy in the form of an electric charge. It consists of two electrical conductors that are separated by a distance. The space between the conductors may be filled by vacuum or with an insulating material known as a dielectric. The ability of the capacitor to store charges is known as capacitance.
Capacitors store energy by holding apart pairs of opposite charges. The simplest design for a capacitor is a parallel plate, which consists of two metal plates with a gap between them. But, different types of capacitors are manufactured in many forms, styles, lengths, girths, and materials.
For proof, let's look at the most basic structure of a capacitor – the parallel plate capacitor. It consists of two parallel plates separated by a dielectric. When we connect a DC voltage source across the capacitor, one plate is connected to the positive end (plate I) and the other to the negative end (plate II). When battery potential is applied across the capacitor, plate I become positive with respect to plate II. The current tries to flow through the capacitor at the steady-state condition from its positive plate to the negative plate. But it cannot flow due to the separation of the plates by insulating material.
An electric field appears across the capacitor. The positive plate (plate I) collects positive charges from the battery and the negative plate (plate II) accumulates negative charges from the battery. After a point, the capacitor holds the maximum amount of is proportional to its voltage times its capacitance. This period is called the capacitor charging time.
When those plates are connected to a load, current flows into the load from plate I to plate II until all the charges are dissipated from both plates. This period is called the time of discharge of the capacitor.
Determine the Value of Capacitance
The conducting plates have some charges Q1 and Q2 (Usually, if one plate has +q, the other has –q charge). The electric field in the region between the plates depends on the charge given to the conducting plates. We also know that potential difference (V) is directly proportional to the electric field hence we can say,
This constant of proportionality is known as the capacitance of the capacitor.
Capacitance is the ratio of the change in the electric charge of a system to the corresponding change in its electric potential.
Energy Stored in a Capacitor
The energy stored in a capacitor is nothing but the electric potential energy and is related to the voltage and charge on the capacitor. If the capacitance of a conductor is C, then it is initially uncharged and it acquires a potential difference V when connected to a battery. If q is the charge on the plate at that time, then
Applications of Capacitor Energy
Standard Units of Capacitance
The basic unit of capacitance is Farad. But, Farad is a large unit for practical tasks. Hence, capacitance is usually measured in the sub-units of Farads, such as micro-farads (µF) or pico-farads (pF).
Most of the electrical and electronic applications are covered by the following standard unit (SI) prefixes for easy calculations:
· 1 mF (millifarad) = 10−3 F
· 1 μF (microfarad) =10−6 F
· 1 nF (nanofarad) = 10−9 F
· 1 pF (picofarad) = 10−12 F
Parallel Plate Capacitor
Parallel Plate Capacitors are formed by an arrangement of electrodes and insulating material or dielectric. A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. It can be defined as:
Parallel Plate Capacitor Formula
The direction of the electric field is defined as the direction in which the positive test charge would flow. Capacitance is the limitation of the body to store the electric charge. Every capacitor has its capacitance. The typical parallel-plate capacitor consists of two metallic plates of area A, separated by the distance d.
The parallel plate capacitor formula is given by:
- ϵo is the permittivity of space (8.854 × 10−12 F/m)
- k is the relative permittivity of dielectric material
- d is the separation between the plates
- A is the area of plates
Parallel Plate Capacitor Derivation
The figure below depicts a parallel plate capacitor. We can see two large plates placed parallel to each other at a small distance d. The distance between the plates is filled with a dielectric medium as shown by the dotted array. The two plates carry an equal and opposite charge.
Here, we see that the first plate carries a charge +Q and the second carries a charge –Q. The area of each of the plates is A and the distance between these two plates is d. The distance d is much smaller than the area of the plates and we can write d<<A, thus the effect of the plates are considered as infinite plane sheets and the electric field generated by them is treated as that equal to the electric field generated by an infinite plane sheet of uniform surface charge density. As the total charge on plate 1 is Q and the area of the plate is A, the surface charge density can be given as
Similarly, for plate 2 with a total charge equal to –Q and area A, the surface charge density can be given as,
We divide the regions around the parallel plate capacitor into three parts, with region 1 being the area left to the first plate, region 2 being the area between the two plates and region 3 being the area to the right of plate 2.
Capacitance of a Spherical Capacitor
Factors Affecting Capacitance
Area of the Plates
The effect of the area of the plate is that the capacitance is directly proportional to the area. The larger the plate area, the more the capacitance value. Mathematically it is given as:
Effect of Dielectric on Capacitance
Let us consider another capacitor with the exact specifications as taken before. Let us insert a dielectric between the plates such that it fully occupies the space between the plates. As the dielectric enters the field between the plates, it gets polarized by the field, and the charges get arranged such that they act as two charged sheets with a surface charge density of σp and – σp, as shown in the figure below.
The net surface charge density then becomes equivalent to ±(σ – σp).